Mixed variational principle in elasticity theory and canonical systems of equations

The paper proposes a modification of the mixed variational principle from which stationarity conditions are derived in the form of a mixed system of equations resolved for the first derivatives of the displacement and stress components acting in a plane perpendicular to one of the coordinate axes. The variational principle allows decreasing the dimension of the problem of elasticity thus reducing the system of equations to a canonical form. The modified mixed principle helps immediately obtain a canonical system of equations for various applied theories. This possibility is demonstrated with the example of the Timoshenko theory of plates __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 55–62, May 2007.     

An approach to solving the stress problem of elasticity

It is shown that three equilibrium equations and three stress compatibility equations are sufficient to solve the stress problem of elasticity __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 7, pp. 23–28, July 2006.     

Research on a systematic methodology for theory of elasticity

IntroductionTraditionalsolutionmethodsforelastictheorygenerallyconformedtotwodifferentthoughts[1～ 12 ] .Onewasbaseduponpointsthatstartfromdifferentialequations.Theotherwasbaseduponwholethatstartsfromenergyprinciples .Inthispaper,theequivalenceisprovedbetweendifferentialformandintegralformofasystematicmethodologyfortheoryofelasticity .Auniformframeworkofthesystematicmethodologyisestablished .Innewsystemitisexpressedintheformsofdifferentialform ,integralformormixedform .Aseriesofdualrelations…     

The more general displacement solutions for the plane elasticity problems

In this paper,the author obtains the more general displacement solutions for theisotropic plane elasticity problems.The general solution obtained in ref.[1 ]is merelythe particular case of this paper,In comparison with ref.[1],the general solutions ofthis paper contain more arbitrary constants.Thus they may satisfy more boundaryconditions.     

Stress solutions to the three-dimensional problem of elasticity

New representations of the stress tensor in the linear theory of elasticity and thermoelasticity are proposed. These representations satisfy the equilibrium equations and the strain compatibility equation. The stress tensor is expressed in terms of a harmonic tensor or a harmonic vector. The second boundary-value problem for an elastic half-space and an elastic layer is solved as an example __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 8, pp. 3–35, August 2006.     

Invariant recording of elasticity theory equations

An invariant (with respect to rotations) formalization of equations of linear and nonlinear elasticity theory is proposed. An equation of state (in the form of a convex generating potential) for various crystallographic systems is written. An algebraic approach is used, which does not require any geometric constructions related to the analysis of symmetry in crystals. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 127–142, September–October, 2008.     

Stress concentration in the moment theory of elasticity

A multimoment theory of elasticity of inhomogeneous media is used to develop algorithms for determining a sequence of essentially independent fundamental parameters associated with the structure, composition, and characteristics of the medium components. The general form of stress concentration factors is established by analyzing specific models of media. It is found that the mean tangential stresses in a fibrous medium with a transverse stress state are asymmetric. As an example, the methods are used to solve specific problems for plates and shells __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 1, pp. 66–76, January 2007.     

On a representation of the solution of the linear elasticity equations

A new representation of the stress tensor in the linear theory of elasticity is proposed. The representation satisfies the equilibrium equations and the compatibility conditions for strains. In this representation, the stress tensor is expressed in terms of a harmonic vector. The second boundary-value problem for an elastic half-space and elastic layer is considered as an example.Translated from Prikladnaya Mekhanika, Vol. 40, No. 11, pp. 85–91, November 2004.This revised version was published online in April 2005 with a corrected cover date.     

Towards multi-scale continuum elasticity theory

We propose a new method of constructing a series of nested quasicontinuum models, which describe linear elastic behavior of crystal lattices at successively smaller scales. The relevant scales are dictated by the interatomic interactions and are not arbitrary. The novelty of the model is in the use of a decomposition of the displacement field into the coarse part and the micro-level corrections. The coarse contribution is the conventional homogenized displacement field used in classical continuum elasticity. The micro-level corrections are sub-continuum fields representing the fine structure of the boundary layers exhibited by the discrete equilibrium configuration. The model is based on a multi-point Padé approximation in the Fourier space of the discrete Green’s function. We systematically compare the new model with the conventional strain gradient model.      

Integral equation methods in plane asymmetric elasticity

The boundary integral equation method is used to solve the interior and exterior Dirichlet, Neumann and mixed problems of plane micropolar elasticity. In the exterior case, a specific far-field pattern for the displacements and microrotation is introduced without which the classical scheme fails to work. Finally, we discuss the direct method and establish a connection with results obtained previously.     

Generalized variational principles on nonlinear theory of elasticity with finite displacements

Two generalized variational principles on nonlinear theory of elasticity with finitedisplacements in which the σ_(ij),e_(i j)and u_i are all three kinds of independent functionsare suggested in this paper.It isproved that these two generalized variational principles areequivalent to each other if the stress-strain relation is satisfied as constraint.Some specialcases,i.e.generalized variational principles on nonlinear theory of elasticity with smalldeformation,on linear theory with finite deformation and on linear theory with smalldeformation together with the corresponding equivalent theorems are also obtained.All ofthem are related to the three kinds of independent variables.     

On the solution of plane boundary-value problems of elasticity in a rectangle

An algorithm for solving plane boundary-value problems of elasticity for a rectangular domain is expounded. The algorithm is based on a complex-valued representation of the general solution to the differential equations of the plane problem and on the use of Lagrange polynomials to satisfy the boundary conditions. The algorithm can quite easily be implemented in a computer program. This is probably the simplest way of solving boundary-value problems of this class __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 97–102, January 2006.     

On the (u, p) problem in the theory of elasticity

A problem of the theory of elasticity is considered for a body with vectors of displacements u and loads p simultaneously defined on one part of the body and with undefined conditions on the remaining part of the body. For a doubly connected domain, where the vectors u and p are set on one of its boundaries (inner or outer), an iterative method based on reduction of the initial problem to a sequence of mixed problems is justified. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 100–103, May–June, 2006.     

A Boundary Value Problem for an Infinite Elastic Strip with a Semi-infinite Crack

In this paper we study a boundary value problem for an infinite elastic strip with a semi-infinite crack. By using the single and double layer potentials this problem is reduced to a singular integral equation, which is uniquely solved in the Hölder spaces by the Fredholm alternative.     

基于厚板理论分析深水域中弹性浮板的水波响应

基于线性水波理论和Mindlin厚板动力学理论,采用Wiener-Hopf 方法,研究了不同水深水面上弹性浮板在不同入射波数水波作用下的动力学响应问题。首先推导了无限深水域中弹性浮板水波响应的解析解,并将本文分析计算结果与采用其他方法(经典薄板理论)得到的计算结果进行了对比和分析;其次,采用本文方法研究了大型浮板在三种入射波数的水波作用下动弯矩幅值的分布情况;最后,根据其他文献的方法计算了不同水深(有限水深)情况下浮板的动响应,并与本文的计算结果进行了对比分析。     

Rational finite element method for elastic bending of reissner plates

IntroductionAsawell-establishedandsophisticatednumericalanalysistechniqueforthedifferentialequahons,thefiniteelementmethodwasoriginatedindependentlybythedomesticandtheforeignmechanicalinvestigatorstheyyearsago.ItfoundforthefirsttimewideapplicationsintheelasticstfUctolproblemsandwiththedevelopmentofcomputerextendedinvaryingdegreestOalmostalltheengineeringdisciplinesandmanyscientificandtechnologicalfields.Thisleadstoadramaticbreakthroughforthecurrentcomputationalmathematics.Themathematicalbasis…     

The Saint-Venant problem of plane bar under an axial force

I.PrefaceAsweknow,theSaint-Venanttheoryisawell-knownanduniversallyacceptedtheory.Butasoneofthegeneraltheoriesofelasticmechanics,itisnotprovedperfectely,althoughitistestedbyexperiments.Infact,onlyforthesimplestproblemofelasticity,itisnoteasytofindanalytics…     

Symplectic eigenfunction expansion theorem for elasticity of rectangular planes with two simply-supported opposite sides

The eigenvalue problem of the Hamiltonian operator associated with plane elasticity problems is investigated.The eigenfunctions of the operator are directly solved with mixed boundary conditions for the displacement and stress in a rectangular region.The completeness of the eigenfunctions is then proved,providing the feasibility of using separation of variables to solve the problems.A general solution is obtained with the symplectic eigenfunction expansion theorem.     

Historical sense in the historians of the theory of elasticity

This paper studies the historical and epistemological approaches followed by historians of the theory of elasticity. The authors chosen are Todhunter and Pearson, Love, Timoshenko, and Truesdell, who are perhaps the most important historians in English language since the second half of the nineteenth century. It is shown that the historical purpose has changed from one historian to the next, ranging from history as a compilation of what is known to criticism of the evolution of elasticity. The path of progress is investigated and again there are significant differences between historians although overall they have a sense of confluence towards some form of truth. The relation between history and theory is identified as a recurrent theme. And the epistemology in each author, although not explicitly stated, is seen to change following the main trends in their times or before. Some possible avenues of research are mentioned towards the end.     

基于平板振动精确化方程求解动应力集中问题

基于文献[8]给出的平板弯曲振动精确化方程,对含圆孔平板中弹性波散射与动应力集中问题进行了研究.文中给出了分别基于Mindlin板与精确化板方程在不同参数下圆孔动弯矩集中系数的数值结果,并对结果进行了对比分析和讨论.结果表明:在较低频率和薄板情况下,基于文献[8]的方程与基于Mindlin板理论得到的动弯矩结果是基本一致的;在较高频率和厚板情况下,基于文献[8]的方程与基于Mindlin板理论的动弯矩结果相差较大,最大值超出可达16%.由于文献[8]给出的平板振动精确化方程是在没有任何工程假设条件下得到的,因此其分析计算结果更精确一些.